A-Level Mathematics May/June 2024 Q2: Express 6x²-9x-16 / 2x²-5x-12 in partial fractions.

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  The given expression is an improper fraction as the degree of the numerator (2) is equal to the degree of the denominator (2). First, we must perform algebraic long division.

  $$\require{enclose} \begin{array}{r} 3 \\ 2x^2-5x-12 \enclose{longdiv}{6x^2-9x-16} \\ \underline{-(6x^2-15x-36)} \\ 6x+20 \\\end{array}$$

  This shows that $\frac{6x^2-9x-16}{2x^2-5x-12} = 3 + \frac{6x+20}{2x^2-5x-12}$.

  Next, we factorise the denominator: $2x^2-5x-12 = (2x+3)(x-4)$.

  Now we express the remainder term in partial fractions: $\frac{6x+20}{(2x+3)(x-4)} \equiv \frac{B}{2x+3} + \frac{C}{x-4}$

  Multiplying by the denominator gives the identity: $6x+20 \equiv B(x-4) + C(2x+3)$

  To find the constants, we substitute values of $x$ that simplify the equation.

  Let $x=4$: $6(4)+20 = B(4-4) + C(2(4)+3)$ $24+20 = 0 + C(8+3)$ $44 = 11C$ $C = 4$

  Let $x = -\frac{3}{2}$: $6(-\frac{3}{2}) + 20 = B(-\frac{3}{2}-4) + C(2(-\frac{3}{2})+3)$ $-9 + 20 = B(-\frac{11}{2}) + 0$ $11 = -\frac{11}{2}B$ $B = -2$

  The constant term from the long division is $A=3$.

  Combining all parts, the final expression is: $3 - \frac{2}{2x+3} + \frac{4}{x-4}$

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  How the marks are awarded

  • B1 — Stating or implying the correct form for an improper fraction, A + B/(2x+3) + C/(x-4). This is achieved by performing the long division to find the constant term A=3 and setting up the remaining fraction for decomposition.
  • M1 — Using a correct method to find a constant for the fractional part. This is awarded for creating the identity 6x+20 ≡ B(x-4) + C(2x+3) and then substituting a value for x (e.g., x=4 or x=-3/2) or equating coefficients.
  • A1 — Obtaining one of the three constants correctly. In this working, A=3 is found first from the long division.
  • A1 — Obtaining a second constant correctly. In this working, C=4 is the second constant found.
  • A1 — Obtaining the third and final constant correctly. In this working, B=-2 is the last constant found, completing the solution.

  Common mistakes

  • Forgetting to perform algebraic long division first, and incorrectly attempting to split the improper fraction directly into the form A/(2x+3) + B/(x-4).
  • Making an arithmetic error during the long division subtraction step, leading to an incorrect remainder (e.g., 6x - 52 instead of 6x + 20), which then leads to incorrect values for the constants B and C.
  • Incorrectly factorising the quadratic denominator, for example as (2x-4)(x+3), which prevents finding the correct partial fractions.
  • Making a sign error when substituting a negative value to find a constant, such as substituting x = -3/2 and calculating B(-11/2) = 11 to incorrectly get B=2 instead of B=-2.

  Examiner tip: Always check if a rational function is improper (degree of numerator ≥ degree of denominator) and perform algebraic long division first before proceeding with partial fractions.

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